Name: unit circle trig: signs and symmetry
Uploaded: 2025-11-26T05:44:36.000Z
Duration: 1 min 30 s
Channel: JoVE

Le cercle unité, centré à l’origine, définit les fonctions trigonométriques.Chaque nombre réel représente une longueur d’arc mesurée à partir de l’axe des x positifs et tourne-le dans le sens inverse des aiguilles d’une montre autour du cercle.La coordonnée x donne le cosinus du nombre réel. De même, la coordonnée y donne le sinus.La tangente est dérivée du sinus sur le cosinus, la cotangente est le cosinus sur le sinus, la sécante est l’inverse du cosinus et la cosécante est l’inverse du sinus.La division du cercle en quatre quadrants permet d’identifier le signe d’une fonction.Toutes les fonctions sont positives dans le premier quadrant, où x et y sont positifs. Dans la seconde, puisque x est négatif, le cosinus est négatif. Comme y est positif, sinus est également positif. La tangente, qui est y sur x, est négative car un positif sur un négatif est négatif.Dans la troisième, la tangente et la cotangente sont positives, comme x et y sont négatives. Dans la quatrième, seuls le cosinus et la sécante sont positifs, car x est positif et y est négatif.Les fonctions trigonométriques sont utiles pour les calculs du monde réel, tels que l’estimation de la hauteur d’un bâtiment en multipliant la distance horizontale connue par la tangente de l’angle d’inclinaison connu.

fonctions trigonométriques des nombres réels

Trigonometric Functions of Real Numbers

Le cercle unit&#233;, un cercle de rayon un, centr&#233; &#224; l&#8217;origine du plan cart&#233;sien, sert de cadre fondamental &#224; la d&#233;finition des fonctions trigonom&#233;triques. Dans ce contexte, la longueur d&#8217;arc d&#233;signe la distance mesur&#233;e le long de la circonf&#233;rence du cercle entre deux points et permet de repr&#233;senter g&#233;om&#233;triquement les nombres r&#233;els. Chaque nombre r&#233;el t correspond &#224; une longueur d&#8217;arc mesur&#233;e dans le sens trigonom&#233;trique &#224; partir de l&#8217;axe des abscisses orient&#233; positivement autour du cercle. Les coordonn&#233;es d&#8217;un point du cercle unit&#233; (x, y) pour une longueur d&#8217;arc donn&#233;e d&#233;finissent respectivement le cosinus et le sinus du nombre r&#233;el, avec x = cos t et y = sin t.D&#8217;autres fonctions trigonom&#233;triques d&#233;rivent directement du sinus et du cosinus. La fonction tangente est d&#233;finie comme le rapport du sinus au cosinus :<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40073/40073_Equation_1_1__editor_40073.svg' alt='40073_Equation_1_1_.svg' /></figure>Inversement, la fonction cotangente est le rapport du cosinus au sinus :<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40073/40073_Equation_2_1__editor_40073.svg' alt='40073_Equation_2_1_.svg' /></figure>Les fonctions s&#233;cante et cos&#233;cante sont respectivement les inverses du cosinus et du sinus&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40073/40073_Equation_3_1__editor_40073.svg' alt='40073_Equation_3_1_.svg' /></figure>La division du cercle unit&#233; en quatre quadrants permet de clarifier les signes des fonctions trigonom&#233;triques. Dans le premier quadrant, toutes les fonctions sont positives. Dans le deuxi&#232;me quadrant, seuls le sinus et la cos&#233;cante conservent des valeurs positives. Dans le troisi&#232;me quadrant, la tangente et la cotangente sont positives, tandis que le cosinus et la s&#233;cante sont positifs dans le quatri&#232;me quadrant. Ces r&#232;gles de signe sont essentielles pour &#233;valuer les fonctions trigonom&#233;triques sous diff&#233;rents angles.La sym&#233;trie du cercle unit&#233; conf&#232;re aux fonctions des propri&#233;t&#233;s de parit&#233;. Le cosinus et la s&#233;cante sont des fonctions paires, v&#233;rifiant cos (&#8722;t) = cos t et sec (&#8722;t) = sec t. En revanche, le sinus, la tangente, la cotangente et la cos&#233;cante sont des fonctions impaires, v&#233;rifiant des identit&#233;s telles que sin (&#8722;t) = &#8722;sin t et tan (&#8722;t) = &#8722;tan t.Ensemble, ces d&#233;finitions et conventions de signe offrent un cadre coh&#233;rent pour comprendre le comportement des fonctions trigonom&#233;triques. Le cercle unit&#233; fournit non seulement une intuition g&#233;om&#233;trique, mais facilite &#233;galement des calculs et des analyses pr&#233;cis, ce qui le rend indispensable en math&#233;matiques pures et appliqu&#233;es.

Fonctions trigonométriques des nombres réels

Le cercle unit&#233;, un cercle de rayon un, centr&#233; &#224; l&#8217;origine du plan cart&#233;sien, sert de cadre fondamental &#224; la ...

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unit circle trig: signs and symmetry

The unit circle is the basis for trigonometric functions on real numbers. It is a circle with radius one centered at the origin of the coordinate plane. On this circle, arc length means the distance measured along the circumference between two points. Each real number t is shown as an arc length measured counterclockwise from the positive x-axis.

A point on the unit circle has coordinates (x, y). These coordinates define cosine and sine. The x-coordinate equals cos t, and the y-coordinate equals sin t. The other trig functions come from these two values. Tangent is the ratio of sine to cosine. Cotangent is the ratio of cosine to sine. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine.

The four quadrants of the unit circle help show where each trigonometric function is positive or negative. In the first quadrant, all trigonometric functions are positive. In the second quadrant, only sine and cosecant are positive. In the third quadrant, tangent and cotangent are positive. In the fourth quadrant, cosine and secant are positive. These sign patterns are important when evaluating trig values for different angles.

The unit circle also shows symmetry in trigonometric functions. Cosine and secant are even functions, so cos(-t) = cos t and sec(-t) = sec t. Sine, tangent, cotangent, and cosecant are odd functions. For example, sin(-t) = -sin t and tan(-t) = -tan t. Together, these rules give a clear way to study trig functions and compute them across the coordinate plane.

Unit Circle Trig: Signs and Symmetry

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trigonometric functions of real numbers

The unit circle is the basis for trigonometric functions on real numbers. It is a circle with radius one centered at the origin of the coordinate plane. On this circle, arc length means the distance measured along the circumference between two points. Each real number t is shown as an arc length measured counterclockwise from the positive x-axis. A point on the unit circle has coordinates (x, y). These coordinates define cosine and sine. The x-coordinate equals cos t, and the y-coordinate...

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Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity. That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second...

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key terms and concepts

Explore how the unit circle defines sine, cosine, tangent, and related functions, plus quadrant signs and even-odd symmetry for real numbers.

Unit Circle Trig: Signs and Symmetry | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Unit Circle Trig: Signs and Symmetry

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